Mortality, Redundancy, and Diversity in Stochastic Search

We investigate a stochastic search process in one dimension under the competing roles of mortality, redundancy, and diversity of the searchers. This picture represents a toy model for the fertilization of an oocyte by sperm. A population of $N$ independent and mortal diffusing searchers all start at $x=L$ and attempt to reach the target at $x=0$. When mortality is irrelevant, the search time scales as ${\tau }_D\mathrm{/}\ln N$ for $\ln N\gg 1$, where ${\tau }_D\sim L^2\mathrm{/}D$ is the diffusive time scale. Conversely, when the mortality rate $\mu$ of the searchers is sufficiently large, the search time scales as $\sqrt{{\tau }_D\mathrm{/}\mu }$, independent of $N$. When searchers have distinct and high mortalities, a subpopulation with a non-trivial optimal diffusivity are most likely to reach the target. We also discuss the effect of chemotaxis on the search time and its fluctuations.